Random event

event

Any combination of outcomes of an experiment that has a definite probability of occurrence.

Example 1. In the throwing of two dice, each of the 36 outcomes can be represented as a pair $( i , j )$, where $i$ is the number of dots on the upper face of the first dice and $j$ the number on the second. The event "the sum of the dots is equal to 11" is just the combination of the two outcomes $( 5 , 6 )$ and $( 6 , 5 )$.

Example 2. In the random throwing of two points into an interval $[ 0 , 1 ]$, the set of all outcomes can be represented as the set of points $( x , y )$( where $x$ is the value of the first point and $y$ that of the second) in the square $\{ {( x , y ) } : {0 \leq x \leq 1, 0 \leq y \leq 1 } \}$. The event "the length of the interval joining x and y is less than a, 0<a< 1" is just the set of points in the square whose distance from the diagonal passing through the origin is less than $\alpha \sqrt 2$.

Within the limits of the generally accepted axiomatics of probability theory (see [1]), where at the base of the probability model lies a probability space $( \Omega , {\mathcal A} , {\mathsf P} )$( $\Omega$ is a space of elementary events, i.e. the set of all possible outcomes of a given experiment, ${\mathcal A}$ is a $\sigma$- algebra of subsets of $\Omega$ and ${\mathsf P}$ is a probability measure defined on ${\mathcal A}$), random events are just the sets which belong to ${\mathcal A}$.

In the first of the above examples, $\Omega$ is a finite set of 36 elements: the pairs $( i , j )$, $1 \leq i , j \leq 6$; ${\mathcal A}$ is the class of all $2 ^ {36}$ subsets of $\Omega$( including $\Omega$ itself and the empty set $\emptyset$), and for every $A \in {\mathcal A}$ the probability ${\mathsf P} ( A)$ is equal to $m / 36$, where $m$ is the number of elements of $A$. In the second example, $\Omega$ is the set of points in the unit square, ${\mathcal A}$ is the class of its Borel subsets and ${\mathsf P}$ is ordinary Lebesgue measure on ${\mathcal A}$( which for simple figures coincides with their area).

The class ${\mathcal A}$ of events associated with $( \Omega , {\mathcal A} , {\mathsf P} )$ forms a Boolean ring with identity with respect to the operations $A + B = ( A \setminus B ) \cup ( B \setminus A )$( symmetric difference) and $A \cdot B = A \cap B$( it has a multiplicative identity $\Omega$), that is, it forms a Boolean algebra. The function ${\mathsf P} ( A)$ defined on this Boolean algebra has all the properties of a norm except one: it does not follow from ${\mathsf P} ( A) = 0$ that $A = \emptyset$. By declaring two events to be equivalent if the ${\mathsf P}$- measure of their symmetric difference is zero, and considering equivalence classes $\overline{A}\;$ instead of events $A$, one obtains the normalized Boolean algebra $\overline {\mathcal A} \;$ of classes $\overline{A}\;$. This observation leads to another possible approach to the axiomatics of probability theory, in which the basic object is not the probability space connected with a given experiment, but a normalized Boolean algebra of random events (see [2], [3]).

References

[1]	A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)
[2]	B.V. Gnedenko, A.N. Kolmogorov, "Probability theory" , Mathematics in the USSR during thirty years: 1917–1947 , Moscow-Leningrad (1948) pp. 701–727 (In Russian)
[3]	A.N. Kolmogorov, "Algèbres de Boole métriques complètes" , VI Zjazd Mathematyków Polskich , Kraków (1950)
[4]	P.R. Halmos, "Measure theory" , v. Nostrand (1950)

References

[a1]	W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1957)
[a2]	H. Bauer, "Probability theory and elements of measure theory", Holt, Rinehart & Winston (1972) pp. Chapt. 11 (Translated from German)